📚 Practical Case Studies for Edexcel Year 11 Further Maths | Edexcel 11年级进阶数学案例分析实战演练
In Edexcel Year 11 Further Maths, applying mathematical concepts to real-world scenarios helps build deeper understanding and exam confidence. Case studies combine topics such as calculus, exponential functions, logarithms, and vectors to solve practical problems. This article presents three detailed case studies with step-by-step analysis, demonstrating how to model, differentiate, integrate, and interpret solutions in context.
在Edexcel 11年级进阶数学中，将数学概念应用于实际场景有助于加深理解和提升考试信心。案例分析将微积分、指数函数、对数和向量等主题结合起来解决实际问题。本文提供三个详细的案例分析，通过逐步分析展示如何在上下文中建立模型、求导、积分并解释结果。
1. Case Study 1: Optimising an Open Box – Setting Up the Problem | 案例1：优化无盖盒体积——问题建立
A manufacturer has a square sheet of cardboard measuring 30 cm by 30 cm. To create an open box, squares of side length x cm are cut from each corner, and the sides are folded up. The goal is to determine the value of x that maximises the volume of the box.
一家制造商有一张边长为30厘米的正方形纸板。为了制作一个无盖盒，从四个角各切去边长为 x 厘米的正方形，然后将四边折起。目标是求出使盒体体积最大的 x 值。
First, express the length, width, and height of the box in terms of x. After removing the corners, the length and width of the base become (30 - 2x) cm, since 2x is removed from each side. The height of the box equals x cm. Therefore, the volume V (in cm³) is given by:
首先，用 x 表示盒子的长、宽和高。切去四角后，底面的长和宽变为 (30 - 2x) 厘米，因为每条边被切去了 2x。盒子的高等于 x 厘米。因此，体积 V（单位为 cm³）为：
V = x (30 - 2x)²
The domain of x is restricted: x must be greater than 0 and less than 15 (since 30 - 2x > 0). This gives a practical domain of 0 < x < 15.
x 的取值范围有限制：x 必须大于 0 且小于 15（因为 30 - 2x > 0）。因此实际定义域为 0 < x < 15。
2. Differentiating to Find Maximum Volume | 求导寻找最大体积
To find the maximum volume, we first expand V into a polynomial to make differentiation straightforward:
为了找到最大体积，我们首先将 V 展开为多项式以便求导：
V = x (900 - 120x + 4x²) = 4x³ - 120x² + 900x
Now differentiate V with respect to x:
现在对 V 关于 x 求导：
dV/dx = 12x² - 240x + 900
Set the derivative equal to zero to locate stationary points:
令导数等于零以确定驻点：
12x² - 240x + 900 = 0
Divide through by 12 to simplify:
方程两边除以 12 进行化简：
x² - 20x + 75 = 0
This quadratic can be solved by factorisation:
该二次方程可通过因式分解求解：
(x - 5)(x - 15) = 0
would have zero base area), so the only meaningful interior stationary point is x = 5 cm.
因此驻点为 x = 5 和 x = 15。然而 x = 15 位于边界（盒子的底面积将为零），所以唯一有意义的内部驻点是 x = 5 厘米。
3. Verifying the Maximum and Concluding | 验证最大值并总结
maximum, examine the second derivative or use a sign test. Compute the second derivative:
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